The sharp p -Poincaré inequality under the measure contraction property
aa r X i v : . [ m a t h . M G ] M a y The sharp p -Poincar´e inequality under the measurecontraction property Bang-Xian Han ∗ May 29, 2019
Abstract
We obtain sharp estimate on p -spectral gaps, or equivalently optimal con-stant in p -Poincar´e inequalities, for metric measure spaces satisfying measurecontraction property. We also prove the rigidity for the sharp p -spectral gap. Keywords : p -Poincar´e inequality, p -spectral gap, p -Obata theorem, curvature-dimension condition, measure contraction property, metric measure space. Contents p -Poincar´e inequalities . . . . . . . . . . . . . . . . . 6 p -spectral gap 11 p -spectral gap estimates . . . . . . . . . . . . . . . . . . . . . . 114.2 Rigidity for p -spectral gap . . . . . . . . . . . . . . . . . . . . . . . . 12 Sharp estimates on spectral gap for p -Laplacian, or equivalently, the optimal con-stant in p -Poincar´e inequalities is a classical problem in comparison geometry. Itaddresses the following basic problem. Given a family F := { ( X α , d α , m α ) : α ∈ A } ∗ Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel.Email: [email protected].
1f metric measure spaces, the corresponding optimal constant λ F in p -Poincar´e in-equalities is defined by λ F := inf α ∈ A inf ( R X α |∇ d α f | p d m α R X α | f | p d m α : f ∈ Lip ∩ L p , Z X α f | f | p − d m α = 0 , f = 0 ) , (1.1)where the local Lipschitz constant |∇ d α f | : X α R is defined by |∇ d α f | ( x ) := lim y → x | f ( y ) − f ( x ) | d α ( y, x ) . One of the most studied families of metric measure spaces is Riemannian mani-folds with lower Ricci curvature bound K ∈ R , upper dimension bound N >
D >
0. In this case, λ F is the minimum of all first positive eigenval-ues of the p -Laplacian (assuming Neumann boundary conditions if the boundary isnot empty). Based on a refined gradient comparison technique and a careful anal-ysis of the underlying model spaces, sharp estimate on the first eigenvalue of the p -Laplacian is finally obtained by Valtorta and Naber in [22, 26].Another important family is weighted Riemannian manifolds (called smooth met-ric measure spaces) satisfying BE( K, N ) curvature-dimension condition `a la Bakry-´Emery [5,6]. More generally, thanks to the deveploment of optimal transport theory,it was realized that Bakry- ´Emery’s condition in smooth setting can be equivalentlycharacterized by convexity of an entropy functional along L -Wasserstein geodesics(c.f. [14] and [27]). In this direction, metric measure spaces satisfying CD( K, N )condition was introduced by Lott-Villani [20] and Sturm [24, 25]. This class of met-ric measure spaces with synthetic lower Ricci curvature bound and upper dimen-sion bound includes the previous smooth examples, and is closed in the measuredGromov-Hausdorff topology. Recently, using measure decomposition technique onRiemannian manifolds developed by Klartag [19] (and by Cavalletti-Mondino [10]on metric measure spaces), sharp p -Poincar´e inequalities under the BE( K, N ) con-dition and the CD(
K, N ) condition have been obtained by E. Calderon in his Ph.Dthesis [9].In addition, Measure Contraction Property MCP(
K, N ) was introduced inde-pendently by Ohta [23] and Sturm [25] as a weaker variant of CD(
K, N ) condition.The family MCP(
K, N ) is strictly larger than CD(
K, N ). It is discovered by Juil-let [18] that the n -th Heisenberg group equipped with the left-invariant measure,which is the simplest sub-Riemannian space, does not satisfy any CD( K, N ) condi-tion but do satisfy MCP(0 , N ) for N ≥ n + 3. Recently, interpolation inequalities`a la Cordero-Erausquin–McCann–Schmuckenshl¨ager [14] have been obtained, undersuitable modifications, by Barilari and Rizzi [8] in the ideal sub-Riemannian setting,Badreddine and Rifford [4] for Lipschitz Carnot group, and by Balogh, Kristly andSipos [7] for the Heisenberg group. As a consequence, more and more examples ofspaces verifying MCP but not CD have been found, e.g. the generalized H-typegroups and the Grushin plane (for more details, see [8]).In [17], the author and E. Milman prove a sharp Poincar´e inequality for subsetsof (essentially non-branching) MCP( K, N ) metric measure spaces, whose diameter isbounded from above by D . The current paper is a subsequent work of [17]. We will2tudy the general p -poincar´e inequality within the class of spaces verifying measurecontraction property. Thanks to measure decomposition theorem (c.f. Theorem 3.5[12]), it suffices to study the corresponding eigenvalue problems on one-dimensionalmodel spaces introduced by E. Milman [21]. In particular, we identify a family ofone-dimensional MCP( K, N )-densities with diameter D , not verifying CD( K, N ),achieving the optimal constant λ pK,N,D .As a basic problem in metric geometry, the rigidity theorem helps us to un-derstand more about the spaces under study. For the family of metric measurespaces satisfying RCD( K, N ) condition with
K >
0, a space reaches the equalityin (1.1) must have maximal diameter π q N − K . By maximal diameter theorem thisspace is isomorphic to a spherical suspension (see [11] and references therein fordetails). For MCP( K, N ) spaces, the situation is very different. For
K >
0, due tolack of monotonicity, we do not know whether a space reaches the minimal spec-trum has maximal diameter. For
K >
0, by monotonicity (Proposition 3.6) andone-dimensional rigidity (Theorem 3.12) we can prove rigidity Theorem 4.2.
Acknowledgement : This research is part of a project which has received fundingfrom the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 637851). The authorthanks Emanuel Milman for helpful discussions and comments.
Let ( X, d) be a complete metric space and m be a locally finite Borel measurewith full support. Denote by Geo(X , d) the space of geodesics. We say that a setΓ ⊂ Geo(X , d) is non-branching if for any γ , γ ∈ Γ, it holds: ∃ t ∈ (0 ,
1) s.t. ∀ s ∈ [0 , t ] γ t = γ t ⇒ γ s = γ s , ∀ s ∈ [0 , . Let ( µ t ) be a L -Wasserstein geodesic. Denote by OptGeo( µ , µ ) the space ofall probability measures Π ∈ P (Geo(X , d)) such that ( e t ) ♯ Π = µ t (c.f. Theorem2.10 [1]) where e t denotes the evaluation map e t ( γ ) := γ t . We say that ( X, d , m )is essentially non-branching if for any µ , µ ≪ m , Π is concentrated on a set ofnon-branching geodesics.It is clear that if ( X, d) is a smooth Riemannian manifold then any subset Γ ⊂ Geo(X , d) is a set of non-branching geodesics, in particular any smooth Riemannianmanifold is essentially non-branching. In addition, many sub-Riemannian spacesare also essentially non-branching, which follows from the existence and uniquenessof the optimal transport map on some ideal sub-Riemannian manifolds (c.f. [15]).Given K, N ∈ R , with N >
1, we set for ( t, θ ) ∈ [0 , × R + , σ ( t ) K,N (cid:0) θ ) := ∞ , if Kθ ≥ ( N − π , sin( tθ √ K/ ( N − θ √ K/ ( N − , if 0 < Kθ < ( N − π ,t, if Kθ = 0 , sinh( tθ √ − K/ ( N − θ √ − K/ ( N − , if Kθ < . τ ( t ) K,N := t N (cid:16) σ ( t ) K,N − (cid:17) − N . Definition 2.1 (Measure Contraction Property MCP(
K, N )) . We say that an es-sentially non-branching metric measure space ( X, d , m ) satisfies measure contrac-tion property MCP( K, N ) if for any point o ∈ supp m and Borel set A ⊂ X with 0 < m ( A ) < ∞ (and with A ⊂ B ( o, p ( N − /K if K > π ∈ OptGeo( m (A) m | A , δ o ) such that the following inequality holds for all t ∈ [0 , m ( A ) m ≥ ( e t ) ♯ (cid:2) τ (1 − t ) K,N (cid:0) d( γ , γ ) (cid:1) N Π(d γ ) (cid:3) . (2.1) Theorem 2.2 (Localization for MCP(
K, N ) spaces, Theorem 3.5 [12]) . Let ( X, d , m ) be an essentially non-branching metric measure space satisfying MCP(
K, N ) con-dition for some K ∈ R and N ∈ (1 , ∞ ) . Then for any 1-Lipschitz function u on X , the non-branching transport set T u associated with u (roughly speaking, T u coincides with {|∇ u | = 1 } up to m -measure zero set) admits a disjoint family ofunparameterized geodesics { X q } q ∈ Q such that m ( T u \ ∪ X q ) = 0 , and m | T u = Z Q m q d q ( q ) , q ( Q ) = 1 and m q ( X q ) = 1 q − a.e. q ∈ Q . Furthermore, for q -a.e. q ∈ Q , m q is a Radon measure with m q ≪ H | X q and ( X q , d , m q ) satisfies MCP(
K, N ) . Let h ∈ L ( R + , L ) be a non-negative Borel function. It is known (see e.g. Lemma4.1 [17]) that (supp h, | · | , h L ) satisfies MCP( K, N ) condition if and only if h is aMCP( K, N ) density in the following sense h ( tx + (1 − t ) x ) ≥ σ (1 − t ) K,N − ( | x − x | ) N − h ( x ) (3.1)for all x , x ∈ supp h and t ∈ [0 , Definition 3.1.
Given K ∈ R , N >
1. Denote by D K,N the Bonnet–Meyers diam-eter upper-bound: D K,N := ( π √ K/ ( N − if K > ∞ otherwise . (3.2)For any D >
0, we define F K,N,D as the collection of MCP(
K, N ) densities h ∈ L ( R + , L ) with supp h = [0 , D ∧ D K,N ]. 4or κ ∈ R , we define the function s κ : [0 , + ∞ ) R (on [0 , π/ √ κ ) if κ > s κ ( θ ) := (1 / √ κ ) sin( √ κθ ) , if κ > ,θ, if κ = 0 , (1 / √− κ ) sinh( √− κθ ) , if κ < . It can be seen that (3.1) is equivalent to (cid:18) s K/ ( N − ( b − x ) s K/ ( N − ( b − x ) (cid:19) N − ≤ h ( x ) h ( x ) ≤ (cid:18) s K/ ( N − ( x − a ) s K/ ( N − ( x − a ) (cid:19) N − (3.3)for all [ x , x ] ⊂ [ a, b ] ⊂ supp h .Furthermore, we have the following characterization. Lemma 3.2.
Given D ≤ D K,N . A density h is in F K,N,D if and only if (cid:18) s K/ ( N − ( D − x ) s K/ ( N − ( D − x ) (cid:19) N − ≤ h ( x ) h ( x ) ≤ (cid:18) s K/ ( N − ( x ) s K/ ( N − ( x ) (cid:19) N − ∀ ≤ x ≤ x ≤ D. (3.4) Furthermore, h ∈ F K,N,D if and only if ln h is L -a.e. differentiable and − h ( x ) cot K,N,D ( D − x ) ≤ h ′ ( x ) ≤ h ( x ) cot K,N,D ( x ) , L − a.e. x ∈ [0 , D ] where the function cot K,N,D : [0 , D ] R is defined by cot K,N,D ( x ) := p K ( N −
1) cot( q KN − x ) , if K > , ( N − /x, if K = 0 , p − K ( N −
1) coth( q − KN − x ) , if K < . Proof.
It can be checked that the function a s K/ ( N − ( x − a ) s K/ ( N − ( x − a )is non-decreasing on [0 , x ], and the function b s K/ ( N − ( b − x ) s K/ ( N − ( b − x )is non-decreasing on [ x , D ]. Thus (3.4) follows from (3.3).Furthermore, for any h ∈ F K,N,D , it can be seen that (3.4) holds if and only if x (cid:0) s K/ ( N − ( D − x ) (cid:1) N − h ( x ) is non-increasing , (3.5)and x (cid:0) s K/ ( N − ( x ) (cid:1) N − h ( x ) is non-decreasing . (3.6)From (3.4) we can see that ln h is locally Lipschitz, so ln h is differentiable almosteverywhere. Thus by (3.5) and (3.6) we know (3.4) is equivalent to (cid:16) ln s N − K/ ( N − ( D − · ) (cid:17) ′ ≤ (ln h ) ′ = h ′ h ≤ (cid:16) ln s N − K/ ( N − (cid:17) ′ L − a.e. on [0 , D ]which is the thesis. 5otice that the function[0 , D ] ∋ x s K/ ( N − ( D − x ) s K/ ( N − ( x )is decreasing. By Lemma 3.2 (or (3.5) and (3.6) ) we immediately obtain the fol-lowing rigidity result. Lemma 3.3 (One dimensional rigidity) . Denote h K,N,D = (cid:0) s K/ ( N − ( x ) (cid:1) N − | [0 ,D ] and h K,N,D = (cid:0) s K/ ( N − ( D − x ) (cid:1) N − | [0 ,D ] . Then we have h K,N,D , h K,N,D ∈ F K,N,D .Furthermore, h K,N,D is the unique F K,N,D density (up to multiplicative constants)satisfying h ′ ( x ) = h ( x ) cot K,N,D ( x ) and h K,N,D is the unique F K,N,D density satisfying h ′ ( x ) = − h ( x ) cot K,N,D ( D − x ) . p -Poincar´e inequalities Definition 3.4.
For p ∈ (1 , ∞ ) and h ∈ F K,N,D , the p -spectral gap associated with h is defined by λ p,h := inf (cid:26) R | u ′ | p h d x R | u | p h d x : u ∈ Lip ∩ L p , Z u | u | p − h d x = 0 , u = 0 (cid:27) . (3.7) Definition 3.5.
Given K ∈ R , D >
N >
1. The optimal constant λ pK,N,D is defined as the infimum of all p -spectral gaps associated with admissible densities,i.e. λ pK,N,D is given by λ pK,N,D := inf h ∈∪ D ′≤ D F K,N,D ′ λ p,h . Proposition 3.6.
Given K ∈ R , D > and N > . The function D λ pK,N,D isnon-increasing, and λ pK,N,D = inf h ∈∪ D ′≤ D F K,N,D ′ ∩ C ∞ λ p,h . (3.8) If K ≤ , the map D λ pK,N,D is strictly decreasing, and λ pK,N,D = inf h ∈ F K,N,D ∩ C ∞ λ p,h . (3.9) Proof.
By Lemma 3.2 we know MCP densities are locally Lipschitz, using a standardmollifier we can approximate h uniformly by smooth MCP densities. Then by asimple approximation argument (see e.g. Proposition 4.8 [17]) we can prove λ pK,N,D = inf h ∈∪ D ′≤ D F K,N,D ′ ∩ C ∞ λ p,h . Let h ∈ F K,N,D ′ be a MCP density for some D ′ >
0, and u be a admissible func-tion in (3.7). Then ¯ h ( x ) := h ( D ′ D x ) ∈ F K ′ ,N,D with K ′ = (cid:0) D ′ D (cid:1) K , and ¯ u ( x ) := u ( D ′ D x )6s also an admissible function. By computation, we have R | ¯ u ′ | p ¯ h d x R | ¯ u | p ¯ h d x = (cid:0) D ′ D (cid:1) p R | u ′ | p h d x R | u | p h d x .Therefore, if K ≤ D ′ < D , we haveinf h ∈ F K,N,D λ p,h ≤ inf h ∈ F K ′ ,N,D λ p,h ≤ (cid:18) D ′ D (cid:19) p (cid:0) inf h ∈ F K,N,D ′ λ p,h (cid:1) < inf h ∈ F K,N,D ′ λ p,h and so λ pK,N,D < λ pK,N,D ′ . Then we obtain (3.9).
Remark . The difference between the cases K ≤ K > false when
K > p and cos p . Definition 3.8.
For p ∈ (1 , + ∞ ), define π p by π p := Z − d t (1 − | t | p ) p = 2 πp sin( π/p ) > . The periodic C function sin p : R [ − ,
1] is defined on [ − π p / , π p /
2] by: ( t = R sin p ( t )0 d s (1 −| s | p ) p if t ∈ [ − π p , π p ] , sin p ( t ) = sin p ( π p − t ) if t ∈ [ π p , π p ] . (3.10)It can be seen that sin p (0) = 0 and sin p is strictly increasing on [ − π p , π p ]. Definecos p ( t ) = dd t sin p ( t ), then we have the following generalized trigonometric identity | sin p ( t ) | p + | cos p ( t ) | p = 1 . Definition 3.9.
Let h iK,N,D , i = 1 , K, N ) densities defined in Lemma3.3. Define h K,N,D by h K,N,D ( x ) := (cid:26) h K,N,D ( x ) if x ∈ [ D , D ] h K,N,D ( x ) if x ∈ [0 , D ] . Define T K,N,D by T K,N,D := (cid:0) ln h K,N,D (cid:1) ′ = (cid:26) cot K,N,D ( x ) if x ∈ [ D , D ] − cot K,N,D ( D − x ) if x ∈ [0 , D ] . By Lemma 3.2 we know h K,N,D is a MCP(
K, N ) density. It can be seen that (c.f.Lemma 3.4 [13]) h K,N,D does not satisfy any forms of CD condition.7 heorem 3.10 (One dimensional p -spectral gap) . Given K ∈ R , N > , D > .Let ˆ λ pK,N,D be the minimal λ such that the following initial value problem has asolution: ϕ ′ = (cid:16) λp − (cid:17) p + p − T K,N,D cos p − p ( ϕ ) sin p ( ϕ ) ,ϕ (0) = − π p , ϕ ( D ) = 0 , ϕ ( D ) = π p . (3.11) Then λ p,h ≥ ˆ λ pK,N,D for any h ∈ F K,N,D .Proof.
Step 1.
Firstly we will show the existence of ˆ λ pK,N,D .By Lemma 3.2 we know T K,N,D ∈ C ∞ ((0 , D ) ∪ ( D , D )) and − cot K,N,D ( D − · ) ≤ T K,N,D ≤ cot K,N,D . Denote T = T K,N,D , and denote by u = u T,λ the (unique)solution of the following equation: (cid:26) (cid:0) u ′ | u ′ | p − (cid:1) ′ + T u ′ | u ′ | p − + λu | u | p − = 0 ,u ( D ) = 0 . (3.12)Next we will study the equation (3.12) using a version of the so-called Pf¨ufertransformation. Define the functions e = e T,λ and ϕ = ϕ T,λ by: α := (cid:16) λp − (cid:17) p , αu = e sin p ( ϕ ) , u ′ = e cos p ( ϕ ) . Differentiating the first equation and substituting by the second one, we get αe cos p ( ϕ ) = e ′ sin p ( ϕ ) + e cos p ( ϕ ) ϕ ′ . By (3.12) we obtain | e cos p ( ϕ ) | p − ( p − (cid:0) e ′ cos p ( ϕ ) − e sin p ( ϕ ) ϕ ′ (cid:1) + T e cos p ( ϕ ) | e cos p ( ϕ ) | p − + λα − p e sin p ( ϕ ) | e sin p ( ϕ ) | p − = 0 . In conclusion, we can see that ϕ, e solve the following equation: ( ϕ ′ = α + p − T | cos p ( ϕ ) | p − cos p ( ϕ ) sin p ( ϕ ) , dd t ln e = e ′ e = − p − T | cos p ( ϕ ) | p . (3.13)Consider the following initial valued problem on (0 , D ) ∪ ( D , D ). (cid:26) ϕ ′ = α + p − T | cos p ( ϕ ) | p − cos p ( ϕ ) sin p ( ϕ ) ,ϕ ( D ) = 0 . (3.14)By Cauchy’s theorem we have the existence, uniqueness and continuous dependenceon the parameters. For any ǫ ∈ (0 , D ), there is α > ϕ ′ ( x ) > π p D − ǫ > x ∈ ( ǫ, D ). So there exists a α ∈ [0 , D ) such that ϕ ( a α ) = − π p . Similarly,there is b α ∈ ( D , D ] such that ϕ ( b α ) = π p . Conversely, assume there is α > ϕ for some a α ∈ [0 , D ) and b α ∈ ( D , D ]: (cid:26) ϕ ′ = α + p − T | cos p ( ϕ ) | p − cos p ( ϕ ) sin p ( ϕ ) ,ϕ ( a α ) = − π p , ϕ ( D ) = 0 , ϕ ( b α ) = π p . (3.15)8hen for any α ′ > α , the following problem also has a solution for some a ′ α ∈ ( a α , D )and b ′ α ∈ ( D , b α ) (cid:26) ϕ ′ = α ′ + p − T | cos p ( ϕ ) | p − cos p ( ϕ ) sin p ( ϕ ) ,ϕ ( a ′ α ) = − π p , ϕ ( D ) = 0 , ϕ ( b ′ α ) = π p . (3.16)Therefore there is a minimal ¯ λ ≥ λ > ¯ λ , there exist ϕ = ϕ T,λ ,0 ≤ a λ < D and D < b λ ≤ D such that ϕ ′ = (cid:16) λp − (cid:17) p + p − T cos p − p ( ϕ ) sin p ( ϕ ) ,ϕ ( a λ ) = − π p , ϕ ( D ) = 0 , ϕ ( b λ ) = π p . (3.17)By continuous dependence on the parameter λ , we know (3.17) has a solution ϕ ∞ for γ = ¯ γ , some a ¯ λ ∈ [0 , D ) and b ¯ λ ∈ ( D , D ]. In particular, ¯ λ > T ( x ) = − T ( D − x ) on [0 , D ], by symmetry and minimality (or domainmonotonicity) of ¯ λ , we have a ¯ λ = 0 and b ¯ λ = D . In particular, there is a minimalˆ λ pK,N,D such that the initial value problem (3.11) has a solution ϕ T K,N,D , ˆ λ pK,N,D . Step 2.
Given h ∈ F K,N,D ∩ C ∞ , we will show that ˆ λ pK,N,D ≤ λ p,h .First of all, by a standard variational argument we can see that λ p,h is the smallestpositive real number such that there exists a non-zero u ∈ W ,p ([0 , D ] , h L ) solvingthe following equation (in weak sense):∆ hp u = − λu | u | p − (3.18)with Neumann boundary condition, where ∆ hp u is the weighted p -Laplacian on([0 , D ] , | · | , h L ):∆ hp u = ∆ p u + u ′ | u ′ | p − (log h ) ′ = (cid:0) u ′ | u ′ | p − (cid:1) ′ + u ′ | u ′ | p − h ′ h . By regularity theory we know u ∈ C ,α ∩ W ,p for some α >
0, and u ∈ C ,α if u ′ = 0. Conversely, for any u solving the Neumann problem (3.18), we have R u | u | p − h d x = 0 and R | u ′ | p h d x = λ R | u | p h d x .Assume by contradiction that λ p,h < ˆ λ pK,N,D . From Step 1 we can see that thereis λ < ˆ λ pK,N,D such that the following equation has a (monotone) solution ϕ = ϕ h ′ h ,λ : ϕ ′ = (cid:16) λp − (cid:17) p + p − h ′ h cos p − p ( ϕ ) sin p ( ϕ ) ,ϕ (0) = − π p , ϕ ( D ) = π p , (3.19)Without loss of generality, we may assume there is a ′ ∈ [ D , D ] such that ϕ h ′ h ,λ ( a ′ ) =0. Suppose there is a point x ∈ [ a ′ , D ) such that ϕ h ′ h ,λ ( x ) = ϕ T K,N,D , ˆ λ pK,N,D ( x ). ByLemma 3.2, we know (cid:0) ϕ h ′ h ,λ (cid:1) ′ ( x ) < (cid:0) ϕ T K,N,D , ˆ λ pK,N,D (cid:1) ′ ( x ) .
9o we have ϕ h ′ h ,λ ( x ) < ϕ T K,N,D , ˆ λ pK,N,D ( x )for all x ∈ ( a ′ , D ], which contradicts to the fact that ϕ h ′ h ,λ ( D ) = ϕ T K,N,D , ˆ λ pK,N,D ( D ) = π p . Combining Proposition 3.6 and Theorem 3.10, we get the following corollaryimmediately. Corollary 3.11.
We have the following sharp p -spectral gap estimates for one di-mensional models: λ pK,N,D = ( ˆ λ pK,N,D if K ≤ D ′ ∈ (0 , min( D,D
K,N )] ˆ λ pK,N,D ′ if K > Theorem 3.12 (One dimensional rigidity) . Given K ≤ , N > and D > .If λ p,h = ˆ λ pK,N,D for some h ∈ F K,N,D . Then h = h K,N,D up to a multiplicativeconstant.Proof.
Assume λ p,h = ˆ λ pK,N,D for some h ∈ F K,N,D . Then there is h n ∈ F K,N,D ∩ C ∞ with h n → h uniformly, and a decreasing sequence ( λ p,h n ) with λ p,h n → ˆ λ pK,N,D , suchthat ϕ n = ϕ h ′ nhn ,λ p,hn solves the following equation: ϕ ′ n = (cid:16) λ p,hn p − (cid:17) p + p − h ′ n h n cos p − p ( ϕ n ) sin p ( ϕ n ) ,ϕ n (0) = − π p , ϕ n ( D ) = π p . (3.20)From Lemma 2.1 we know { ϕ ′ n } n and { ϕ n } n are uniformly bounded. By Arzel`a-Ascoli theorem we may assume ϕ n → ϕ ∞ uniformly for some Lipschitz function ϕ ∞ .By minimality of ˆ λ pK,N,D and symmetry, we can see that lim n →∞ ϕ − n ( t ) exists forany t ∈ [ − π p , π p ] and lim n →∞ ϕ − n = (cid:0) ϕ T K,N,D , ˆ λ pK,N,D (cid:1) − . In fact, assume by contradiction that lim n →∞ ϕ − n ( t ) = (cid:0) ϕ T K,N,D , ˆ λ pK,N,D (cid:1) − ( t ) for some t ∈ ( − π p , π p ). By symmetry we may assume there are N ∈ N and δ >
0, such that δ n := (cid:0) ϕ T K,N,D , ˆ λ pK,N,D (cid:1) − ( t ) − ϕ − n ( t ) ≥ δ for all n ≥ N . Define a MCP( K, N ) density¯ h n by ¯ h n ( x ) := ( h n ( x ) if x ∈ [0 , ϕ − n ( t )] , h n ( ϕ − n ( t )) h K,N,D ( ϕ − n ( t )+ δ n ) h K,N,D ( x + δ n ) if x ∈ [ ϕ − n ( t ) , D − δ n ] . Then ¯ ϕ n = ϕ ¯ h ′ n ¯ hn ,λ p,hn satisfies ( ¯ ϕ n ) − ( π p ) < D − δ for n large enough, which contractsto Proposition 3.6 and the minimality of ˆ λ pK,N,D .10n conclusion, ϕ ∞ = ϕ T K,N,D , ˆ λ pK,N,D and we have ϕ n → ϕ T K,N,D , ˆ λ pK,N,D uniformly.Then we get π p ϕ n ( ϕ − n (0)) − ϕ n (0)= lim n →∞ Z ϕ − n (0) ∧ D (cid:16) λ p,h n p − (cid:17) p + 1 p − h ′ n h n cos p − p ( ϕ n ) sin p ( ϕ n ) d x ≤ lim n →∞ Z ϕ − n (0) ∧ D (cid:16) λ p,h n p − (cid:17) p + 1 p − T K,N,D cos p − p ( ϕ n ) sin p ( ϕ n ) d x = Z D (cid:16) ˆ λ pK,N,D p − (cid:17) p + 1 p − T K,N,D cos p − p ( ϕ T K,N,D , ˆ λ pK,N,D ) sin p ( ϕ T K,N,D , ˆ λ pK,N,D ) d x = π p . Therefore (ln h n ) ′ = h ′ n h n → T K,N,D = h ′ K,N,D h K,N,D in L ([0 , D ] , cos p − p ( ϕ T K,N,D , ˆ λ pK,N,D ) sin p ( ϕ T K,N,D , ˆ λ pK,N,D ) L ). By symmetry, we can seethat (ln h n ) ′ → (ln h K,N,D ) ′ in L ([0 , D ] , L ). Hence h = h K,N,D up to a multiplica-tive constant. p -spectral gap p -spectral gap estimates Using standard localization argument (c.f. Theorem 1.1 [17], Theorem 4.4 [11]), wecan prove the sharp p -Poincar´e inequality with one dimensional results. Theorem 4.1 (The sharp p -spectral gap under MCP( K, N )) . Let ( X, d , m ) be anessentially non-branching metric measure space satisfying MCP(
K, N ) for some K ∈ R , N ∈ (1 , ∞ ) and diam( X ) ≤ D . For any p > , define λ p ( X, d , m ) as the optimalconstant in p -Poincar´e inequality on ( X, d , m ) : λ p ( X, d , m ) := inf (cid:26) R |∇ f | p d m R | f | p d m : f ∈ Lip ∩ L p , Z f | f | p − d m = 0 , f = 0 (cid:27) . Then we have the following sharp estimate λ p ( X, d , m ) ≥ λ pK,N,D = ( ˆ λ pK,N,D if K ≤ D ′ ∈ (0 , min( D,D
K,N )] ˆ λ pK,N,D ′ if K > . Proof.
Let ¯ f = f | f | p − be a Lipschitz function with R ¯ f = 0. Let ¯ f ± denote thepositive and the negative parts of ¯ f respectively. Then we have R ¯ f + = − R ¯ f − .Consider the L -optimal transport problem from µ := ¯ f + m to µ := − ¯ f − m . By11heorem 2.2, there exists a family of disjoint unparameterized geodesics { X q } q ∈ Q oflength at most D , such that m ( X \ ∪ X q ) = 0 , m = Z Q m q d q ( q )where m q = h q H | X q for some h q ∈ F K,N,D ′ q with D ′ q ≤ D , m q ( X q ) = m ( X ) and Z ¯ f h q d H | X q = 0for q -a.e. q ∈ Q .Denote f q = f | X q . By definition we obtain Z | f ′ q | p h q d H | X q ≥ λ p,h q Z | f q | p h q d H | X q ≥ λ pK,N,D Z | f q | p h q d H | X q . Notice that | f ′ q | ≤ |∇ f | , we have λ pK,N,D Z | f | p d m = λ pK,N,D Z Q Z X q | f q | p m q d q ( q ) ≤ Z Q Z X q | f ′ q | p m q d q ( q )= Z |∇ f | p d m . Combining with Corollary 3.11 we prove the theorem. p -spectral gap In this part, we will study the rigidity for p -spectral gap under the measure contrac-tion property. We adopt the notation | D f | to denote the weak upper gradient of aSobolev function f . We refer the readers to [2] and [16] for details about Sobolevspace theory and calculus on metric measure spaces. Theorem 4.2 (Rigidity for p -spectral gap) . Let ( X, d , m ) be an essentially non-branching metric measure space satisfying MCP(
K, N ) for some K ≤ , N ∈ (1 , ∞ ) and diam( X ) ≤ D . Assume there is a non-zero Sobolev function f ∈ W ,p ( X, d , m ) with R f | f | p − d m = 0 such that Z | D f | p d m − ˆ λ pK,N,D Z | f | p d m = 0 . Then diam( X ) = D and there are disjoint unparameterized geodesics { X q } q ∈ Q oflength D such that m ( X \ ∪ X q ) = 0 . Moreover, m has the following representation m = Z Q h q d H | X q d q ( q ) , where h ′ q h q = T pK,N,D for q -a.e. q ∈ Q . roof. Similar to the proof of Theorem 4.1, we can find a measure decompositionassociated with ¯ f := f | f | p − , such that m ( X \ ∪ X q ) = 0 , m = Z Q m q d q ( q )where m q = h q H | X q for some h q ∈ F K,N,D ′ q with D ′ q ≤ D , m q ( X q ) = m ( X ) and Z ¯ f h q d H | X q = 0for q -a.e. q ∈ Q .By Theorem 7.3 [3] we know f q := f | X q ∈ W ,q ( X q ) and | D f q | ≤ | D f | . Thenfrom the proof of Theorem 4.1 we can see that λ p,h q = ˆ λ pK,N,D for q -a.e. q ∈ Q .By Proposition 3.6 we know D ˆ λ pK,N,D is strictly decreasing, so D ′ q = D anddiam( X ) = D . Finally, by Theorem 3.12 we know h ′ q h q = T pK,N,D . References [1]
L. Ambrosio and N. Gigli , A user’s guide to optimal transport . Modellingand Optimisation of Flows on Networks, Lecture Notes in Mathematics, Vol.2062, Springer, 2011.[2]
L. Ambrosio, N. Gigli, and G. Savar´e , Calculus and heat flow in metricmeasure spaces and applications to spaces with Ricci bounds from below , Invent.math., (2013), pp. 1–103.[3] ,
Density of Lipschitz functions and equivalence of weak gradients in metricmeasure spaces , Revista Matem´atica Iberoamericana, 29 (2013), pp. 969–996.[4]
Z. Badreddine and L. Rifford , Measure contraction properties for two-step analytic sub-Riemannian structures and Lipschitz carnot groups . Preprint,arXiv:1712.09900, 2017.[5]
D. Bakry , L’hypercontractivit´e et son utilisation en th´eorie des semigroupes ,in Lectures on probability theory (Saint-Flour, 1992), vol. 1581 of Lecture Notesin Math., Springer, Berlin, 1994, pp. 1–114.[6]
D. Bakry and M. ´Emery , Diffusions hypercontractives , in S´eminaire de prob-abilit´es, XIX, 1983/84, vol. 1123 of Lecture Notes in Math., Springer, Berlin,1985, pp. 177–206.[7]
Z. M. Balogh, A. Krist´aly, and K. Sipos , Geometric inequalities onHeisenberg groups , Calc. Var. Partial Differential Equations, 57 (2018), pp. Art.61, 41.[8]
D. Barilari and L. Rizzi , Sub-Riemannian interpolation inequalities , Invent.math., (2018). doi:10.1007/s00222-018-0840-y.139]
E. Calderon , Functional inequalities on weighted Riemannian manifoldssubject to curvature-dimension conditions . Ph.D Thesis, Technion - I.I.T.,arXiv:1905.08866, 2019.[10]
F. Cavalletti and A. Mondino , Sharp and rigid isoperimetric inequalitiesin metric-measure spaces with lower Ricci curvature bounds , Invent. Math., 208(2017), pp. 803–849.[11] ,
Sharp geometric and functional inequalities in metric measure spaces withlower Ricci curvature bounds , Geom. Topol., 21 (2017), pp. 603–645.[12]
F. Cavalletti and A. Mondino , New formulas for the Laplacian of distancefunctions and applications . Preprint, arXiv:1803.09687, 2018.[13]
F. Cavalletti and F. Santarcangelo , Isoperimetric inequality undermeasure-contraction property . Preprint, arXiv:1810.11289, 2018.[14]
D. Cordero-Erausquin, R. J. McCann, and M. Schmuckenschl¨ager , A Riemannian interpolation inequality `a la Borell, Brascamp and Lieb , Invent.Math., 146 (2001), pp. 219–257.[15]
A. Figalli and L. Rifford , Mass transportation on sub-Riemannian man-ifolds , Geom. Funct. Anal., 20 (2010), pp. 124–159.[16]
N. Gigli , On the differential structure of metric measure spaces and applica-tions , Mem. Amer. Math. Soc., 236 (2015), pp. vi+91.[17]
B.-X. Han and E. Milman , The sharp Poincar´e inequality under measurecontraction properties . Preprint, arXiv:1905.05465, 2019.[18]
N. Juillet , Geometric inequalities and generalized Ricci bounds in the Heisen-berg group , Int. Math. Res. Not. IMRN, (2009), pp. 2347–2373.[19]
B. Klartag , Needle decompositions in Riemannian geometry , Mem. Amer.Math. Soc., 249 (2017), pp. v + 77.[20]
J. Lott and C. Villani , Ricci curvature for metric-measure spaces via op-timal transport , Ann. of Math. (2), 169 (2009), pp. 903–991.[21]
E. Milman , Sharp isoperimetric inequalities and model spaces for thecurvature-dimension-diameter condition , J. Eur. Math. Soc. (JEMS), 17 (2015),pp. 1041–1078.[22]
A. Naber and D. Valtorta , Sharp estimates on the first eigenvalue of the p -Laplacian with negative Ricci lower bound , Math. Z., 277 (2014), pp. 867–891.[23] S.-i. Ohta , On the measure contraction property of metric measure spaces ,Comment. Math. Helv., 82 (2007), pp. 805–828.[24]
K.-T. Sturm , On the geometry of metric measure spaces. I , Acta Math., 196(2006), pp. 65–131. 1425] ,
On the geometry of metric measure spaces. II , Acta Math., 196 (2006),pp. 133–177.[26]